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Applications of computer holography
Published in Tomoyoshi Shimobaba, Tomoyoshi Ito, Computer Holography, 2019
Tomoyoshi Shimobaba, Tomoyoshi Ito
The sinc function sinc(x)=sin(x)∕x becomes 0 when x=π. Therefore, the diffraction efficiency η becomes smallest when the argument of the sinc function of Eq. (5.31) is π, that is, () πtsin(θr−θo)Δθλcosθo=π.
Optical fiber devices
Published in John P. Dakin, Robert G. W. Brown, Handbook of Optoelectronics, 2017
Suzanne Lacroix, Xavier Daxhelet
Apodization: For most applications, side lobes seen in the spectral response are undesirable. They originate from the multiple reflections that take place between the grating ends as in a Fabry–Perot resonator. One can also understand this effect by remembering that, in a first approximation (i.e., when the perturbation is weak), the spectral response of a grating is the Fourier transform of the amplitude c of the refractive index modulation. The Fourier transform of a rectangular function is a sinc function. The larger the rectangular function, the narrower the sinc function. Similar observations are valid for the grating: the longer the grating, the narrower the spectral width and its side lobes.
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
The sinc function is defined by sinc(x)=sinxx for x ≠ 0 and sinc(0) = 1. Figure 2 shows the graph of sinc(x) for −5 ≤ x ≤ 5, illustrating the local behavior near 0. Figure 3 shows the graph of sinc(x) for − 100 ≤ x ≤ 100, illustrating the global behavior but distorting the behavior near 0. Note that sinc is an even function. Note also that sinc is complex analytic everywhere in ℂ, because the removable singularity at 0 has been removed! This function also satisfies the following infinite product formulas: sinc(x)=sin(x)x=∏n=1∞(1−x2π2n2).sinc(x)=∏j=1∞cos(x2j
Quantum dynamics of a polar rotor acted upon by an electric rectangular pulse of variable duration
Published in Molecular Physics, 2021
Mallikarjun Karra, Burkhard Schmidt, Bretislav Friedrich
By substituting into Equation (13), models (11) and (12) render, respectively, the following expressions for the coefficients of the initially unpopulated states, where the argument of the sinc function, , is defined in Table 1; it is proportional to the difference of the two eigenvalues – and thus analogous to the factor appearing in the perturbative treatment mentioned above. The zeroes of the sinc function (which coincides with the zeroth-order spherical Bessel function of the first kind, ) occur when the argument ξ is an integer multiple of π. Hence for integer n and a pulse strength P, the real-valued roots of the equations yield, respectively, the values of the pulse durations at which the and coefficients vanish. The first-order Taylor expansions of these pulse durations, included in Equations (16) and (17), indicate that the dependence on P of the loci of the zeros of the and coefficients is parabolic. In the weak-perturbation limit, , the coefficients and have periods π and , respectively.